Explanation:
the surface area is the sum of the base area (pentagon) and the 5 side triangles (we only need to calculate one and then multiply by 5, as they are all equal).
these side triangles are isoceles triangles (the legs are equally long).
the usual area formula for a pentagon is
1/2 × perimeter × apothem
the apothem is the minimum distance from the center of the pentagon to each of its sides.
in our case this is 7.2 mm.
how to get the perimeter or the length of an individual side of the pentagon ?
if the apothem of a pentagon is given, the side length can be calculated with the formula
side length = 2 × apothem length × tan(180/n)
where 'n' is the number of sides (5 in our case). After getting the side length, the perimeter of the pentagon can be calculated with the formula
perimeter = 5 × side length.
so, in our case
side length = 2 × 7.2 × tan(180/5) = 14.4 × tan(36) =
= 10.4622124... mm
perimeter = 5 × 10.4622124... = 52.31106202... mm
area of the pentagon = 1/2 × perimeter × apothem =
= 1/2 × 52.31106202... × 7.2 = 188.3198233... mm²
now for the side triangles.
the area of such a triangle is
1/2 × baseline × height
baseline = pentagon side length
height we get via Pythagoras from the inner pyramid height and the apothem :
height² = 7.2² + 15.4² = 51.84 + 273.16 = 289
height = 17 mm
area of one side triangle =
1/2 × 10.4622124... × 17 = 88.92880543... mm²
all 5 side triangles are then
444.6440271... mm²
and the total surface area is then
444.6440271... + 188.3198233... = 632.9638504... mm²
≈ 632.96 mm²